A261049 -id:A261049 - cp's OEIS frontend

A300508 Expansion of Product_{k>=1} (1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

1, -1, -2, -1, -1, 3, 3, 9, 9, 10, 8, -1, -21, -45, -77, -130, -163, -198, -179, -108, 101, 451, 1058, 1878, 2999, 4276, 5595, 6511, 6446, 4443, -838, -11069, -28373, -54652, -91948, -140370, -198501, -259706, -311997, -332003, -285486, -118600, 239086, 881998, 1918851, 3470261

Offset: 0

Ilya Gutkovskiy, Mar 07 2018

sign

Convolution inverse of A001970.

Alois P. Heinz, Table of n, a(n) for n = 0..3000

Cf. A000041, A001970, A261049.

with(numtheory): with(combinat):
b:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*b(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i)*a(i), i=0..n-1))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 07 2018

nmax = 45; CoefficientList[Series[Product[(1 - x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k)^A000041(k).

A387136 Number of ways to choose a sequence of distinct prime factors, one of each prime index of 2n - 1.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 1, 2, 0, 1, 1, 1, 2, 2, 0, 1, 2, 0, 1, 1, 1, 2, 1, 1, 1, 1, 0, 2, 1, 0, 2, 1, 1, 3, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 2, 2, 2, 0, 2, 2, 0, 1, 1, 0, 1, 2, 1, 2, 2, 0, 2

Offset: 1

Gus Wiseman, Aug 30 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 4537 are {6,70}, with choices (2,5), (2,7), (3,2), (3,5), (3,7). Since 4537 = 2 * 2269 - 1, we have a(2269) = 5.

Here we use the version with alternating zeros (put n instead of 2n - 1 in the name).
Twice partitions of this type are counted by A296122.
Positions of zero are A355529, complement A368100.
For divisors instead of prime factors we have A355739.
Allowing repeated choices gives A355741.
For partitions instead of prime factors we have A387110.
For initial intervals instead of prime factors we have A387111.
For strict partitions instead of prime factors we have A387115, disjoint case A383706.
For constant partitions instead of prime factors we have A387120.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A261049, A276078, A276079, A299200, A335433, A335448, A355731, A355745, A367771, A387133, A387135, A387327.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[If[#==1,{},First/@FactorInteger[#]]&/@prix[2n-1]],UnsameQ@@#&]],{n,100}]

A358904 Number of finite sets of compositions with all equal sums and total sum n.

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 38, 64, 156, 260, 632, 1024, 2601, 4096, 10208, 16944, 40966, 65536, 168672, 262144, 656980, 1090240, 2620928, 4194304, 10862100, 16781584, 41940992, 69872384, 168403448, 268435456, 693528552, 1073741824, 2695006177, 4473400320, 10737385472

Offset: 0

Gus Wiseman, Dec 13 2022

nonn

			The a(1) = 1 through a(4) = 9 sets:
  {(1)}  {(2)}   {(3)}    {(4)}
         {(11)}  {(12)}   {(13)}
                 {(21)}   {(22)}
                 {(111)}  {(31)}
                          {(112)}
                          {(121)}
                          {(211)}
                          {(1111)}
                          {(2),(11)}

This is the constant-sum case of A098407, ordered A358907.
The version for distinct sums is A304961, ordered A336127.
Allowing repetition gives A305552, ordered A074854.
The case of sets of partitions is A359041.
A001970 counts multisets of partitions.
A034691 counts multisets of compositions, ordered A133494.
A261049 counts sets of partitions, ordered A358906.
Cf. A000009, A063834, A075900, A218482, A296122.

Table[If[n==0,1,Sum[Binomial[2^(d-1),n/d],{d,Divisors[n]}]],{n,0,30}]

a(n) = if (n, sumdiv(n, d, binomial(2^(d-1), n/d)), 1); \\ Michel Marcus, Dec 14 2022

a(n>0) = Sum_{d|n} binomial(2^(d-1),n/d).

A358913 Number of finite sequences of distinct sets with total sum n.

Original entry on oeis.org

1, 1, 1, 4, 6, 11, 28, 45, 86, 172, 344, 608, 1135, 2206, 4006, 7689, 13748, 25502, 47406, 86838, 157560, 286642, 522089, 941356, 1718622, 3079218, 5525805, 9902996, 17788396, 31742616, 56694704, 100720516, 178468026, 317019140, 560079704, 991061957

Offset: 0

Gus Wiseman, Dec 11 2022

nonn

			The a(1) = 1 through a(5) = 11 sequences of sets:
  ({1})  ({2})  ({3})      ({4})        ({5})
                ({1,2})    ({1,3})      ({1,4})
                ({1},{2})  ({1},{3})    ({2,3})
                ({2},{1})  ({3},{1})    ({1},{4})
                           ({1},{1,2})  ({2},{3})
                           ({1,2},{1})  ({3},{2})
                                        ({4},{1})
                                        ({1},{1,3})
                                        ({1,2},{2})
                                        ({1,3},{1})
                                        ({2},{1,2})

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The unordered version is A050342, non-strict A261049.
The case of strictly decreasing sums is A279785.
This is the distinct case of A304969.
The case of distinct sums is A336343, constant sums A279791.
This is the case of A358906 with strict partitions.
The version for compositions instead of strict partitions is A358907.
The case of twice-partitions is A358914.
A001970 counts multiset partitions of integer partitions.
A055887 counts sequences of partitions.
A063834 counts twice-partitions.
A330462 counts set systems by total sum and length.
A358830 counts twice-partitions with distinct lengths.
Cf. A000009, A000041, A000219, A271619, A296122, A336342, A358908.

g:= proc(n) option remember; `if`(n=0, 1, add(g(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(binomial(g(i), j)*b(n-i*j, i-1, p+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 13 2024

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp],{comp,Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n],UnsameQ@@#&&And@@UnsameQ@@@#&]],{n,0,10}]

a(n) = Sum_{k} A330462(n,k) * k!.

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 10, 13, 17, 21, 27, 34, 42, 53, 65, 80, 98, 119, 146, 177, 213, 258, 309, 370, 443, 528, 628, 745, 882, 1043, 1229, 1447, 1700, 1993, 2333, 2727, 3182, 3707, 4311, 5008, 5808, 6727, 7782, 8990, 10371, 11952, 13756, 15815, 18161

Offset: 0

Gus Wiseman, Sep 02 2025

First differs from A052337 in having 745 instead of 746.
We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct strict integer partitions of each part.
Also the number of integer partitions of n with no part k whose multiplicity exceeds A000009(k).

			The partition y = (3,3,2) has sets of strict integer partitions ({(2,1),(3)},{(2,1),(3)},{(2)}), and we have the choice ((2,1),(3),(2)) or ((3),(2,1),(2)), so y is counted under a(8).
The a(1) = 1 through a(9) = 10 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,3,1)  (4,4,1)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
                                                            (3,3,2,1)

For initial intervals instead of strict partitions we have A238873, ranks A387112.
For divisors instead of strict partitions we have A239312, ranks A368110.
The complement for divisors is A370320, ranks A355740.
For prime factors instead of strict partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement for initial intervals is A387118, ranks A387113.
The complement for all partitions is A387134, ranks A387577.
The complement is counted by A387137, ranks A387176.
These partitions are ranked by A387177.
For all partitions instead of just strict partitions we have A387328, ranks A387576.
The complement for constant partitions is A387329, ranks A387180.
For constant partitions instead of strict partitions we have A387330, ranks A387181.
A000041 counts integer partitions, strict A000009.
A358914 counts twice-partitions into distinct strict partitions.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A367867, A367901, A387115.

strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Select[Tuples[strptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]

A317535 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} 1/(1 - x^k)).

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 23, 48, 106, 227, 494, 1065, 2310, 4991, 10808, 23376, 50593, 109455, 236858, 512479, 1108924, 2399418, 5191853, 11233929, 24307777, 52596430, 113806948, 246252376, 532834797, 1152933975, 2494689316, 5397944266, 11679933875, 25272740480, 54684508281, 118324934647

Offset: 0

Ilya Gutkovskiy, Jul 30 2018

nonn

Invert transform of A000065.

Robert Israel, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A000041, A000065, A001970, A055887, A063834, A261049, A271619, A304966, A317536.

seq(coeff(series(1/(1+1/(1-x)-mul(1/(1-x^k),k=1..n)), x,n+1),x,n),n=0..40); # Muniru A Asiru, Jul 30 2018

nmax = 35; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[1/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[1/(1 - Sum[(PartitionsP[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(PartitionsP[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 35}]

G.f.: 1/(1 - Sum_{k>=1} A000065(k)*x^k).

A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31

Offset: 0

Gus Wiseman, Oct 12 2018

			The a(1) = 1 through a(10) = 13 clutters:
  {{1}}  {{2}}  {{3}}    {{4}}    {{5}}    {{6}}      {{7}}
                {{1,2}}  {{1,3}}  {{1,4}}  {{1,5}}    {{1,6}}
                                  {{2,3}}  {{2,4}}    {{2,5}}
                                           {{1,2,3}}  {{3,4}}
                                                      {{1,2,4}}
                                                      {{1,2},{1,3}}
.
  {{8}}          {{9}}          {{10}}
  {{1,7}}        {{1,8}}        {{1,9}}
  {{2,6}}        {{2,7}}        {{2,8}}
  {{3,5}}        {{3,6}}        {{3,7}}
  {{1,2,5}}      {{4,5}}        {{4,6}}
  {{1,3,4}}      {{1,2,6}}      {{1,2,7}}
  {{1,2},{1,4}}  {{1,3,5}}      {{1,3,6}}
  {{1,2},{2,3}}  {{2,3,4}}      {{1,4,5}}
                 {{1,2},{1,5}}  {{2,3,5}}
                 {{1,2},{2,4}}  {{1,2,3,4}}
                 {{1,3},{1,4}}  {{1,2},{1,6}}
                 {{1,3},{2,3}}  {{1,2},{2,5}}
                                {{1,3},{1,5}}

Cf. A001970, A007718, A048143, A050342, A056156, A089259, A261049, A293994, A319719, A320351, A320353, A320355, A320356.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 8, 1, 3, 3, 9, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 5, 8, 1, 12, 1, 19, 3, 3, 3, 24, 1, 3, 3, 20, 1, 12, 1, 8, 8, 3, 1, 46, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1, 38, 1, 3, 8, 37, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 46, 9, 3

Offset: 1

Gus Wiseman, Apr 26 2025

nonn

			The a(36) = 24 choices:
  {{2,2,3,3}}  {{2},{2,3,3}}  {{2},{3},{2,3}}
  {{2,2,9}}    {{3},{2,2,3}}  {{2},{3},{6}}
  {{2,3,6}}    {{2,2},{3,3}}
  {{2,18}}     {{2},{2,9}}
  {{3,3,4}}    {{9},{2,2}}
  {{3,12}}     {{2},{3,6}}
  {{4,9}}      {{3},{2,6}}
  {{6,6}}      {{6},{2,3}}
  {{36}}       {{2},{18}}
               {{3},{3,4}}
               {{4},{3,3}}
               {{3},{12}}
               {{4},{9}}

The case of a unique choice (positions of 1) is A008578.
This is the strict case of A050336.
For distinct strict blocks we have A050345.
For integer partitions we have A261049, strict case of A001970.
For strict blocks that are not necessarily distinct we have A296119.
Twice-partitions of this type are counted by A296122.
For normal multisets we have A317776, strict case of A255906.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, distinct A050326.
A281113 counts twice-factorizations, strict A296121, see A296118, A296120.
Cf. A000009, A005117, A045782, A050342, A279785, A293243, A293511, A302494, A316439, A358914, A381992, A382201.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Sum[Length[Select[mps[y],UnsameQ@@#&]],{y,facs[n]}],{n,30}]

A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 2, 6, 16, 40, 96, 226, 512, 1140, 2488, 5336, 11270, 23494, 48356, 98438, 198338, 395846, 783136, 1536800, 2992818, 5786952, 11114950, 21213906, 40247696, 75928804, 142475644, 265985628, 494155176, 913802164, 1682338192, 3084101744, 5630853218, 10240484332, 18553818210

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

nonn

Convolution of the sequences A001970 and A261049.

N. J. A. Sloane, Transforms

Cf. A000041, A001970, A015128, A156616, A206622, A206623, A206624, A260916, A261049, A261386, A261452, A261519, A261520, A301554, A301555, A302237, A302238.

nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A000041(k).

A304784 Expansion of Product_{k>=1} 1/(1 + x^k)^p(k), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, -1, -1, -2, 0, -1, 2, 3, 11, 8, 19, 13, 22, -5, -10, -80, -105, -246, -303, -502, -506, -681, -400, -231, 873, 1956, 4733, 7536, 12891, 17609, 25188, 29508, 34890, 29690, 19039, -17742, -74002, -183563, -333665, -572271, -866683, -1271429, -1698491, -2181207

Offset: 0

Ilya Gutkovskiy, May 18 2018

sign

Convolution inverse of A261049.

Eric Weisstein's World of Mathematics, Partition Function P
Index entries for sequences related to partitions

Cf. A000041, A001970, A089254, A261049, A300508.

nmax = 43; CoefficientList[Series[Product[1/(1 + x^k)^PartitionsP[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) d PartitionsP[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

G.f.: Product_{k>=1} 1/(1 + x^k)^A000041(k).

cp's OEIS Frontend

A300508 Expansion of Product_{k>=1} (1 - x^k)^p(k), where p(k) = number of partitions of k (A000041).

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A387136 Number of ways to choose a sequence of distinct prime factors, one of each prime index of 2n - 1.

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A358904 Number of finite sets of compositions with all equal sums and total sum n.

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A358913 Number of finite sequences of distinct sets with total sum n.

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A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

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A317535 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} 1/(1 - x^k)).

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A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

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A383310 Number of ways to choose a strict multiset partition of a factorization of n into factors > 1.

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A302239 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^p(k), where p(k) = number of partitions of k (A000041).

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